The details of the Argonath's origins given in this entry are taken from Appendix A (iv) to The Lord of the Rings, which states of Rómendacil II: 'He it was that built the pillars of the Argonath at the entrance to Nen Hithoel.' Rómendacil was King of Gondor between the years III 1304 and III 1366, but before taking the throne he had served as Regent since III 1240, and the context of this quote strongly suggests that Pillars were raised during his Regency period, in about the year III 1250.
To complicate matters, all of this seems to be flatly contradicted by The Silmarillion. In Of the Rings of Power and the Third Age, the raising of the Pillars is apparently described as taking place long before, during the last years of the Second Age: '...works marvellous and strong they built in the land in the days of their power, at the Argonath...' This is probably a simple error (and as the ultimate canonical source, the account in The Lord of the Rings must take precedence).
To resolve this apparent contradiction, we might imagine that the 'marvellous works' raised in Elendil's time weren't necessarily the Pillars of the Argonath themselves - perhaps they refer to the long narrow channel that led into Nen Hithoel, with the mighty statues at the entrance to the channel being raised more than a thousand years later by Rómendacil. It's very doubtful that Tolkien intended any of this, but it does help to reconcile the contradictory accounts.